Monopolistic Competition and New Products: A
Conjectural Equilibrium Approach
This version: June 16th, 2009
Francesco Bogliacino and Giorgio Rampa
Universidad EAFIT, Medellin, [email protected]
University of Pavia, [email protected]
Abstract
In this paper we generalize the heterogeneous risk adverse agents model of diffusion of
new products in a multi-firm, heterogeneous and interacting agents environment. We use
a model of choice under uncertainty based on Bayesian theory. We discuss the possibility
of product failures, the set of equilibria, their stability and some welfare properties
depending on the degree of diversification.
Keywords: Product diffusion, Risk aversion, Lock-in, Monopolistic competition, Multiple
equilibria.
JEL Classification System: L15, D81, O33
Corresponding author: Francesco Bogliacino, departamento de Economía, Universidad
EAFIT- Carrera 49, 7 Sur 50-05001000 Medellin (Colombia)- Tel. (+57)42619334- Fax.
(+57)42619294
Acknowledgments: The authors want to thank all the participants to the Winter WEHIA
Workshop in Kainan University – Kaoyuan (Taiwan). A special thanks to Prof Akira
Namatame and an anonymous referee. All the remaining errors are ours.
1
1. Introduction
The analysis of diffusion processes is interesting under at least two different perspectives.
First of all, scholars usually concentrate on new products, but it is possible to generalize
many conclusions to the adoption of new technologies, behaviours, fashions and strategies
(in the game-theoretic sense), so enlarging the focus significantly. Second, diffusion is in
essence a multi-disciplinary matter: the literature that has studied the problem spans from
management to sociology, from psychology to physics including, obviously, alternative
economic approaches1.
The literature has discussed both the conditions that favour or hamper diffusion −bringing
eventually to failure or success− and the speed of diffusion, looking at the factors giving
rise to different possible patterns, and in particular to an epidemiologic-like S-shaped
curve.
A satisfactory picture should be grounded on some essential building blocks. The first one
is uncertainty: the very novelty of goods (ideas, technologies, behaviours etc.) implies
that agents must act using conjectures over some unknown feature, as in standard
Bayesian approaches (Jensen 1982, Feder and O’Mara 1982, Tsur et al. 1990, Chatterjee
and Eliashberg 1990, Young 2006). The second block is heterogeneity: individual models
are necessarily different at the outset, since they summarize personal conjectures,
previous learning and a priori ideas (Abrahamson and Rosenkopf, 1993; Cowan and
Jonard 2003 and 2004; Lopez Pintado and Watts, 2006). The third block is interaction:
the learning activity on the part of agents exploits past observations, stemming mainly
from other agents’ choices. Interaction thus shapes the overall process, making it path
dependent. Coupling all this with some degree of non-linearity might finally allow for
multiple equilibria, and hence non-uniqueness of outcomes (lock-in: see Arthur 1994,
Amable 1992, Agliardi 1998, Aoki and Yoshikawa 2002, Young 2007).
In Bogliacino and Rampa (2008) we developed a setup which includes risk aversion and
the interaction between demand for and supply of a single new product. The presence of
both aspects distances that paper from other existing models. Risk aversion is relevant,
because during the learning process the emergence of information shapes the confidence
of agents (as captured by individual precisions), so altering their willingness to pay. This
aspect was already recognised by some2, but never worked out in a fully fledged
dynamical learning model, as done instead in Bogliacino and Rampa (2008). Demandsupply interaction allows one to free the analysis from the single-sided approach
1
The milestone for the literature on diffusion is the Bass model of epidemiologic diffusion pattern (Bass,
1969). There is a sociological strand of literature focussed on heterogeneity and social effects, e.g.
Granovetter (1978), Macy (1991), Abrahamson and Rosenkopf (1993), Valente (1996), Lopez-Pintado and
Watts (2006). The orthodox Economics literature is more interested in grounding the choice process on
robust roots, using Bayesian theory (Jensen 1982, Feder and O’Mara 1982, Birkhchandani et al. 1992,
Bergemann and Välimäki, 1997 Vettas, 1998), but some discussion on more general behaviour rules can be
found in Nelson et al. (2002) and Geroski (2000). An excellent review is Hall (2005); an overall discussion
of the properties of diffusion curves under alternative setups is Young (2007). On the physics side, one
should consider percolation theory as a model of diffusion of ideas and innovations in networks: see e.g.
Grimmett (1999) and, as an economic application, Duffie and Manso (2006); an econophysics example is
offered by Yanagita and Onozaki (2008).
2
Roberts and Urban (1988).
2
prevailing in the literature3; in addition, this allows to model explicitly firms’ uncertainty
over demand. The main results proved in that paper were: a positive probability of failure
of an otherwise ‘good’ product, depending on consumers’ priors; the possibility of
different shapes of the diffusion curve of the new product (besides the usual S-shaped
one), depending again on consumers’ priors; the existence of a continuum of market
equilibria, depending on the firm’s prior on demand; the possibility that a part of those
equilibria are dynamically unstable under learning, especially if the firm is highly
uncertain about demand; and finally a monotonic relation between the degree of stability
and welfare effects.
In the present paper we generalize the previous results analyzing a multiple good case, i.e.
abandoning monopoly and moving to monopolistic competition, but still retaining
consumers’ and firms’ heterogeneity and uncertainty, and consumers’ risk aversion. As in
the first paper, we provide purely analytical results, characterizing the full set of equilibria
of the diffusion process together with their stability properties under the learning
dynamics, without relying on simple simulations exercises which in the end give only a
partial understanding of the overall process. To our knowledge this is the first article that
analyzes analytically the property of a model with multiple goods, two-side uncertainty
and heterogeneous agents. Moreover, in the end of the paper we endogenize the number
of firms, i.e. we discuss entry, while in Proposition 1 we discuss failure, i.e. exit. This
means that this paper is launching the basis for an overall treatment of the establishment
of a new market under heterogeneity and interaction, without relying simply on
simulations.
The paper proceeds as follows: Section 2 describes the demand side, Section 3 the supply
one, Section 4 presents and discusses the main results, Section 5 concludes. Proofs are
collected in the Appendix.
2. Consumers
The individual consumer j ( j = 1,K M ) maximizes her expected utility choosing the level
of consumption of each new good i ( i = 1,K n ), over whose qualities she is uncertain.
Qualities are independent normal variables, with known precision and unknown mean.
Following a standard Bayesian setting, we assume consumers to be endowed with a prior
over the unknown mean quality of each good, defined by two hyper-parameters µ j , i ,t and
τ j ,i , t , respectively the mean and the precision (the inverse of the variance, see DeGroot,
1970) that evolve through time being updated using Bayes’ rule. We assume additively
separable preferences. From now on t denotes time, ranging discretely from zero onwards.
We represent the consumer problem in the following way:
n
max E[U ( x j ,i ,t , λi ) | µ j ,i ,t −1 ,τ j ,i ,t −1 ] = E[∑ u ( x j ,i ,t ) f (λi ) | µ j ,i ,t −1 ,τ j ,i ,t −1 ]
{ x j ,i ,t }i =1,K, n
i =1
n
such that
∑p
i ,t
x j ,i ,t ≤ w
i =1
3
Some noteworthy exceptions are Bergemann and Välimäki (1997) and Vettas (1998).
3
(1)
where pi ,t is the price of good i at time t, and w is the income endowment, for simplicity
equal in time and through all consumers4. U is an additively separable utility, where u is
the sub-utility5 function (i.e. the utility of the single good), and x j ,i ,t the quantity of good i
consumed by agent j at time t. The function f (⋅) is the way the quality of each good ( λi )
is incorporated into agents’ preferences.
In particular, as in the single good framework of Bogliacino and Rampa (2008), we
∂ 2U
assume that U satisfies (i)
> 0 , meaning that the consumer wishes to purchase
∂x j ,i ,t ∂λi
∂ 3U
< 0 , i.e. consumers are risk
2
∂x j ,i , t (∂λi )
averse in quality6: this suggests that a higher variance of quality tends to depress
(expected) marginal utility and hence consumption, for given price.
more if quality is higher, for given price; and (ii)
As in Bogliacino and Rampa (2008), we posit u(⋅) = (⋅)δ , and f ( λi ) = A − exp(−λi ); we
assume in addition λi ~ N(µi ,r), due to random production and/or delivery factors, where
the true mean µi is unknown, and r is known, to consumers; the different qualities are
statistically independent. The individual prior, defined over the mean of each quality, is
also assumed normal, which allows us to use the properties of the conjugate family7. The
advantage of these assumptions is threefold: first, they satisfy the two conditions (i-ii)
above; second, they allow us to “pass through” the expected value operator using the fact
that, owing to normality and to the exponential, f ( λi ) is log-normal; finally, they imply,
as we shall see, that consumers are not bound to buy a positive quantity of each good.
This last property is useful to study the effects of noisy quality signals on consumers’
choices, addressing the possibility of lock-in, i.e. the failure of a diffusion of a “good”
product.
As regards the timing of events, the consumer makes her choice at time t using all
information available at that time, which is captured through her posterior, and before
knowing the others’ choices at t. All the new information refers then to choices made at
t−1, hence the hyper-parameters relevant for the choice at t are µ j ,i ,t −1 and τ j , i , t −1 .
This given, standard maximization8 implies the following individual demand curve for
each good j:
4
In (1) agents take expectations with respect to all the available information at time t, which obviously
includes the information revealed by the market in the previous period, thus we use the time subscript t-1
for hyper-parameters. The reason will become clear in a while.
5
The term sub-utility is standard in the literature that uses additively separable structure, either summing
through time of through goods. In our setup it simply expresses the utility of the consumption of one
specific good.
6
In the standard choice theory, risk aversion is deemed as negativity of the second derivative. In our setup,
this property obviously holds for quality, since u(⋅) is strictly increasing and the utility function U is
multiplicatively separable in quality and quantity. However, we preferred to present this characteristic in
terms of third cross-derivative, because we want to stress the implication for the quantity purchased.
7
A utility function similar to that used in the present setup was proposed also by Roberts and Urban (1988),
who however did not explore analytically the dynamic implications of learning and of demand-supply
interaction, limiting themselves to simulations exercises.
8
We refer to the solution of the problem (1), i.e. maximization of expected utility under a budget constraint,
where the choice variables are the quantities of each good, and the functional forms of u and f are as said
above. The problem is strictly concave on a compact feasible set, which implies that the solution is unique.
4
1/(1−δ )
 

τ −1 j ,i ,t −1 + r −1  


 
pi ,t
A
−
exp
−
+
δ
µ
 
j ,i ,t −1


2
 

 
=
w
1 /(1−δ )
−1
−1
n






r
τ
+
j ,i ,t −1
δ /( δ −1)
 
pi ,t
δ  A − exp − µ j ,i ,t −1 +
∑

2
i =1
 

 
1/( δ −1)
x j ,i ,t
(2)
where one must intend x j , i ,t = 0 whenever A − exp(⋅) ≤ 0 9. If A − exp(⋅) > 0 , we say that
consumer j is active on market i at time t.
The interpretation is straightforward: each consumer spends a share of its total income on
good i, depending on the ratio of its price-quality term to that of the whole bundle of
goods. Total actual market demand for good i, QiD,t , is simply the summation over the j
index.
After buying the chosen quantity, each active consumer receives a quality signal that she
publicly announces to all consumers: these signals are used by each of them to update her
conjecture. Using the properties of conjugate families (DeGroot, 1970), the posterior
parameters for the normal-normal couple (respectively, the likelihood and the prior) are
calculated simply as:
µ j ,i ,t =
τ j ,i ,t −1µ j ,i ,t −1 + rM i ,t λi ,t
, τ j ,i ,t = τ j ,i ,t −1 + rM i ,t
τ j ,i ,t −1 + rM i ,t
(3)
where λi,t is the quality sample mean, computed from the announced perceived qualities,
and M i ,t ≤ M is the number of active buyers at date t. Notice that consumers treat
qualities as independent, and update their conjectures accordingly (that is, separately for
each good).
The above equation simply tells us that consumers average their own prior opinions and
the sample mean of quality from the new observations, the weight being the relative
precisions of the two measures. Moreover, through time individual precisions grow
linearly: as one can imagine, given the assumptions of quality-risk-aversion, this fact
tends to raise demand in time, due to a simple informational effect.
3. Firms
Firms interact in monopolistic competition, each producing a new good at a constant
marginal cost c i : since each firm corresponds to a single product, as in standard
monopolistic competition, we use the i index to define a firm. Every firm is uncertain over
its own demand. To make things as simple as possible, we assume that it conjectures a
linear demand defined by two parameters: more precisely, given the price pi ,t , firm i
believes that its demand is a random normal variable with mean Qi ,t = ai − bi pi ,t and
precision equal to 1. In addition, firm i does not know ai and bi , and maintains the
δ
In fact, although the sub-utility u(⋅) = (⋅) satisfies Inada conditions, when this condition holds, the perperiod utility becomes negative, except if the quantity is zero: thus not buying becomes the rational choice.
9
5
hypothesis that the distribution of the two parameters is a normal bivariate: the mean and
the precision hyper-parameters of this distribution at date t are as follows10:
 αi ,t 
m i ,t =   ,
 
 βi ,t 
 γi ,1,t
Γ i ,t = 

γi , 21,t
γi ,12,t 


γi , 2,t 
(4)
where γi ,1,t and γi , 2,t are positive. Since the firm has surely no reason to conjecture any
particular initial value for the correlation among the two mean hyper-parameters, we
assume γ i ,12, 0 = γ i , 21, 0 = 0 . Define also γ i ,k ≡ γ i ,k , 0 , k = 1,2 as the firm’s initial precisions of
the mean parameters.
As in the consumer case, the timing is as follows: the firm announces the price before
observing demand, hence it uses its (t−1)-conjecture, formed observing demand at time
t−1. The firm chooses the price so as to maximize expected profit. Therefore, from
standard First Order Condition in monopoly, the price announced at date t is:
pi,t =
α i,t−1 c i
+
2β i,t−1 2
(5)
and expects the following demand:
Qie,t ( pi ,t ) =
α i ,t −1 − ci β i ,t −1
(6)
2
We neglect any capacity constraint, and assume that the firm can meet all demand11.
The updating process on the part of firm i follows, again, standard Bayesian rules: using
primes to denote transposed vectors, define the row vector x'i ,t ≡ [1 − pi ,t ] . Given our
assumptions, one has (DeGroot, 1970, Chapter 11):
m i ,t = [Γ i ,t −1 + x i ,t −1x′i ,t −1 ]−1[Γ i ,t −1m i ,t −1 + x i ,t −1QiD,t ]
(7)
and
Γ i ,t = [Γ i ,t −1 + x i ,t −1x′i ,t −1 ]
(8)
By simple algebra, (7) can be rewritten as:
10
This derives from our assumption that the conditional distribution of Qi ,t has known precision equal to 1;
if this precision were different from 1, the precision matrix Γ i,t would be multiplied by its value. Things
could be generalized, but this would be immaterial for our results, since firm’s expected profit does not
depend on precisions, given risk neutrality.
11
An interesting aspect of our setup is the possibility of analysing disequilibrium processes leaving its main
features unaltered. In fact, assuming for instance production lags, i.e. the need for the firms to decide
quantity and price, then equilibrium, as defined short below, is also a market equilibrium in the standard
sense. From (9) it is clear that all that is needed to discuss the disequilibrium path and the convergence to a
market equilibrium is the possibility for firms to observe the true demand for given price in each period and
to adjust supply accordingly, e.g. through the use of inventories. However this full characterization is
beyond the scope of the present paper, being a question more related to discussion of the features of a
general equilibrium.
6
m i ,t = [Γ i ,t −1 + x i ,t −1x′i ,t −1 ]−1[(Γ i ,t −1 + x i ,t −1x′i ,t −1 − x i ,t −1x′i ,t −1 )m i ,t −1 + x i ,t −1QiD,t ] =
= [Γ i ,t −1 + x i ,t −1x′i ,t −1 ]−1[(Γ i ,t −1 + x i ,t −1x′i ,t −1 )m i ,t −1 + xi ,t −1 (QiD,t − x′i ,t −1m i ,t −1 )] =
(9)
= m i ,t −1 + [Γ i ,t −1 + x i ,t −1x′i ,t −1 ]−1[x i ,t −1 (QiD,t − x′i ,t −1m i ,t −1 )]
In a nutshell, the above expression tells us that the new mean parameters are equal to the
previous period’s ones, plus a correction term depending the prediction error12 and
adjusted for the new precision matrix.
4. Equilibria: Main Results
The system can be fully characterized in terms of firms’ and consumers’ hyperparameters.
Define µ j , t = [ µ j ,1, t ... µ j , n, t ]′ and τ j , t = [τ j ,1,t ...τ j , n,t ]′ as the vectors of consumer j’s hyperparameters at time t. Then define µ t = [µ1′,t ... µ′M ,t ]′
and τ t = [ τ1′,t ... τ′M ,t ]′ for all
consumers. As regards firms, call γ i ,t = [γ i ,1,t γ i ,12,t γ i , 21,t γ i , 2,t ]′ the vectorization of the
precision matrix of firm i’s conjecture at time t; posit finally γ t = [ γ1′,t ... γ′n,t ]′ , and
m t = [m1′,t ... m′n ,t ]′ .
Defining y t = [µ′t τ′t m′t γ′t ]' , we compact all the updating equations13 in the following
system of 2nM + 6n first order difference equations:
y t = F (y t −1 )
(10)
which completely describes the learning and diffusion dynamics.
Risk aversion on the part of consumers makes them sensitive to all piece of information
available: as time goes by, new information can increase precisions and raise their
demand, ceteris paribus. For this reason the system shows path dependence and
irreversibility. The relevant equilibrium concept is thus a steady state one, meaning the
agents’ conjectures remain fixed in time. We use in fact a conjectural equilibrium notion:
a conjectural equilibrium is a fixed point of (10).
One might think that a conjectural equilibrium requires that all consumers have
necessarily learnt the true qualities of the goods. In fact, if new information keeps
arriving, the Law of Large Numbers implies that consumers are bound to learn the true
qualities. It is also possible, however, that consumers are endowed initially with
pessimistic conjectures about one of the goods, so demanding a null quantity of it: a null
demand, in turn, implies that no signal will arrive at next date, and conjectures remain
unchanged (lock-in). More importantly, it might happen that, even starting from a positive
demand at date t, a highly biased signal switches demand off at date t + 1 : we term
“failure” this phenomenon.
As regards this last point, we recall one of the results of Bogliacino and Rampa (2008).
Proposition 1. Suppose that demand for good i is positive at time t. Then, there exists
positive probability of failure of the i-th product at time t + 1.
12
13
Notice in fact that x′i ,t −1m i ,t −1 is expected demand, given the prior.
Taking account of (2) and (5).
7
Proof. See Bogliacino and Rampa (2008), Proposition 1.
The argument runs as follows: at every time t we can build a complete ordering over the
set of consumers in terms of a function of their mean and precision hyper-parameters (the
indicator being τ ijt −1 ( µijt −1 − τ −1ijt −1 / 2 − B) , where B = r −1 − log( A) ): the higher its value,
the higher a consumer’s ‘optimism’. If a signal is so biased as to drive the most optimistic
consumer below a certain threshold (recall that A < exp(−λi ) implies no purchase), then
all demands are driven to zero. But then no information is made available to update
conjectures, and consumers are locked-in at zero demand14. Finally, using our
assumptions on distributions, we can prove that the probability of such an event is always
positive. We do not report the complete proof here, since it is quite long and in addition
the mentioned ordering has a number of interesting implications that are worth
investigating on their own.
We come now to a different set of results, assuming that failure does not occur. In this
case a conjectural equilibrium is a situation in which consumers’ conjectured means have
converged to the true mean qualities, and in addition firms’ conjectures are confirmed by
the true demands, so that prediction errors are zero and firms’ conjectures remain
unchanged at subsequent dates15. We can fix the ideas taking µi , j ,t = µi , ∀i, j , and
studying the dynamics in expected value terms16, i.e. with the signals always equal to the
true qualities, so that demands stay constant for given prices (and consumers’ precisions
are free to diverge as in the standard Bayesian setting).
This given, define
g i (m i ,t −1 ) = QiD,t [ pi ,t (m i ,t −1 ), µi ] − x'i ,t m i ,t −1
as the excess of actual demand over expected one for good i; thus the equilibrium
condition can be written as follows:
g (m t ) = [ g1 (m1,t ) ... g n (m n,t )]' = 0 ,
(11)
a set of n equations. Then the following Proposition holds.
Proposition 2. There exists a n-dimensional equilibrium manifold in the space of firms’
parameters.
Proof. (11) is a system of n equations in 2n variables, hence the set of solutions is
generically a n-dimensional manifold.
Conjectural equilibria, then, form a continuum: there is not a unique steady state that can
be attained by the system. Although mathematically simple, this proposition states that the
reliance on “fundamentals equilibrium reasoning” by standard theory, leading to
uniqueness results, should be taken with caution, since it is not robust to the weakening of
the common prior assumption. Notice that different conjectural equilibria differ among
14
One can also, as in Bogliacino and Rampa (2008), study the diffusion dynamics and microfound logistic
or concave diffusion patterns depending on initial consumer conjectures, an aim which is beyond the scope
of this paper. This question is a key one in the literature on diffusion, looking for an explanation of
aggregate specific patterns of diffusion through time based on individual choice, and not single-equation
aggregate behaviour as in the case of Bass (1969). In Bogliacino and Rampa (2008) we show how the
hyper-parameters can be used to characterize individual conjectures giving rise to the different results.
15
See expression (9) above.
16
With respect to the true distribution of µi.
8
themselves as regards prices and quantities, and hence welfare as well, not as regards the
qualities perceived by consumers, since we are assuming that these have been learned
perfectly.
A natural question is now the stability of equilibria along the manifold. Given the
continuum, we must speak of Lyapunov stability: that is, stable equilibria are not
asymptotically (locally) stable, since a small displacement from one stable equilibrium to
another does not cause convergence back to the former. In addition, in the case of
stability, different initial conditions lead to different final states.
We study stability of equilibria at any finite time, recalling that we are assuming
µi , j ,t = µi , ∀i, j and are working in expected values. Hence the stability of equilibria
depends entirely on the firms’ parameters: indeed we can prove the following Proposition.
Proposition 3. The equilibria where conjectured demand is more elastic than the true one
are locally unstable.
Proof. See Appendix A.1.
Figure 1. An equilibrium where conjectured demand is more elastic than the true one.
The intuition for this result can be seen using Figure 1. A possible equilibrium position is
A, where the firm maximizes profits, given its conjecture, and there is no prediction error.
From the definition of equilibrium, price and quantity are common to both the true and
the conjectured demand, so the condition of Proposition 3 implies that the derivative of
the conjectured demand is higher (in absolute value) than that of true demand. On the
contrary, a B-like equilibrium is one where the true demand is less rigid than the
conjectured one.
Look at expression (9), and at how it can be rewritten according to Appendix A.2: in the
presence of excess demand a firm updates its parameters in such a way that the α
9
parameter grows and the β parameter decreases17. Hence, using (5), it follows that firm
will raise its price at the subsequent date. The opposite holds in the presence of excess
supply.
Consider now what is happening in a neighbourhood of A; a higher (resp. smaller) price,
such as p h (resp. p l ) generates excess supply (resp. demand), thus inducing the firm to
raising (resp. lowering) further its price. It is then apparent that the system moves away
from the A equilibrium. A similar reasoning for a B-like equilibrium shows that in this
case there can exist a basin of attraction (unless there is overshooting, a possibility shown
by Proposition 4 below). This type of instability is obviously local, since we can only
study linear approximations.
In a B-like equilibrium we could still observe local instability at some finite time,
instability being of the oscillatory type. This property, however, is smoothed by the
passing of time and the instability is rapidly reabsorbed. In fact we have the following
Proposition.
Proposition 4. In an equilibrium where the true elasticity is high and the demand
conjectured by a single firm is more rigid than the true one, there can exist oscillatory
instability as long as t is small, and provided that the firms’ initial precisions are low.
Proof. See Appendix A.2.
Under the condition of this Proposition, if the system starts in a neighbourhood of some
equilibrium the variables will be pushed away from it, and, given the continuum of
equilibria, the location of the steady state depends on initial conditions. Observe however
that the same unstable equilibria are turned into stable ones by the passing of time, that
has the effect of increasing firms’ precisions, as apparent from the proof of Proposition 4.
We can finally add some further results in terms of welfare. In Bogliacino and Rampa
(2008), studying a single firm, we analyzed the relation between welfare and stability
along the equilibrium manifold. In the present context the higher dimensionality makes
things more complex: it is not so easy to identify how individual parameters change
together along the manifold; and we cannot block n-1 firms, trying to concentrate on a
single one, since changing one price implies obviously changes in all expenditure shares.
We leave this point for further research.
Our multiple-good setup, however, allows us to analyze the degree of diversification of
the decentralized economy and its welfare properties, although under some stricter
assumptions. This is a fairly standard procedure in Monopolistic Competition literature:
we need to endogenize the number of firms (i.e. the number of varieties) by means of a
fixed cost of entry (see Dixit and Stiglitz, 1977; Tirole, 1988; Bertoletti et al. 2008), then
17
In A.1 it is shown that (9) is equivalent to m i ,t = m i ,t −1 + C i ,t g i (⋅) , where Ci , t
 γ i,2
+ tpi ,t

= d i ,t  pi ,t

t


tpi , t 
,
γ i ,1 + t 
d i ,t > 0 , g i (⋅) ≡ [g i (⋅) − g i (⋅)]′ , and g i (⋅) was defined before expression (11) above: see (16) and (23) in
that Appendix. As a consequence, one can easily check that if g i (⋅) > 0 , that is, if true demand exceeds
conjectured demand, then the first element of C i ,t g i (⋅) is positive, while the second is negative.
10
free entry implies a zero profit condition, which closes the model. Indeed, the following
Proposition holds.
Proposition 5. In equilibrium with endogenous number of firms (assuming a positive
fixed cost of entry) and identical marginal cost and qualities of goods, there is over (resp.
under) diversification, if for the marginal firms −defined as that who fix the price at the
lowest level in equilibrium− the true elasticity is greater (resp. lower) than the
conjectured one.
Proof. See Appendix A.3.
The interpretation is fairly obvious. Define ε T and ε C to be the true elasticity and that
conjectured by firms: when the full information case is characterised by efficiency, in
εT
> 1 makes firms less able to appropriate surplus, since it
εC
ε
pushes entry. The opposite holds for T < 1 . Since the full information case is efficient,
εC
equilibrium the condition
when conjectured and true elasticity are equal the actual degree of diversification is equal
to the optimal one. Thus, interestingly, not only the case
εT
= 1 is stable, as implied by
εC
Propositions 3 and 4 taken together: it is also efficient in terms of diversification.
A caveat about this result: it is partly dependent on the particular form of the utility
function. In general, the relation between the optimal degree of diversification and that
prevailing under perfect information depends on how consumers’ preferences affect the
mark-up, since the latter is related to the ability of firms to appropriate the surplus (see
Dixit and Stiglitz 1977). In our case, the iso-elastic assumption guarantees efficiency.
However in the general case the ratio among the true elasticity and the conjectured one
still allows us to characterize over and under diversification with respect to the perfect
information case; of course one cannot say any longer that the degree of diversification
prevailing under perfect information is also optimal.
5. Conclusions
This work studied a monopolistic competitive market, where firms innovate introducing
new products and are uncertain about demand; at the same time, consumers are
heterogeneous as regards their expectations on product qualities, which they are uncertain
about. A key feature of the setup is quality risk aversion on the part of consumers,
affecting their willingness to pay for products, due to their degree of uncertainty that in
turn depends on past choices of all agents. Indeed, there is interaction in time among and
between the market sides: such interaction shapes the learning process and the final
pattern observed. In spite of the seeming complexity, we are able to characterize
analytically some relevant properties of the stationary states of the dynamics without
resorting to simulations, as is instead common in many studies of product diffusion.
The main results can be summarised as follows. First, there is positive probability of lockin, that is high-quality products can fail to diffuse while lower-quality ones can succeed:
11
this does not depend on some ‘objective’ increasing returns (as in, e.g., Arthur, 1984), but
on the constellation of consumers’ priors, coupled with learning and risk aversion.
Second, differently from the “fundamentals equilibrium reasoning” of standard theory, we
find a continuum of conjectural equilibria, i.e. stationary states of the learning process.
These two results are common to the single-firm case studied by Bogliacino and Rampa
(2008): however, in the present monopolistic-competition setup the topological dimension
of the equilibrium manifold is higher, since it equals the number of firms or goods; in
other terms, we have a higher degree of indeterminacy.
The multi-firm case analysed in this work shows in addition that the (local) stability
properties of conjectural equilibria under the learning dynamics can be multifarious: if a
firm conjectures a demand curve that is more elastic than the true one, then we have
monotonically unstable equilibria (Proposition 3). However, it is not always the case that,
if conjectured demands are all less elastic than true ones, then equilibria are stable:
indeed, if the true elasticity is high, and if a single firm conjectures a low elasticity but at
the same time is very uncertain, then we could observe unstable oscillations around
equilibria (Proposition 4). These unstable equilibria, that in view of the subsequent
Proposition 5 are somehow inefficient, turn into stable ones when firms’ uncertainty
decreases, i.e. as time elapses. This means that it is not true that learning or ‘evolution’
weeds out inefficiency.
Finally, the setup has been used to study endogenous entry and optimal diversification, at
least under some further assumptions. We proved that the case when firms conjecture an
elasticity equal to the true one in equilibrium is not only dynamically stable, but is also
efficient in terms of diversification. In our decentralised environment, however, the
learning process can approach any one of a very large number of stationary states, not
only that efficient result, depending on firms’ priors about demand. This observation
makes room for possible corrective policies. In addition, also the above-mentioned lock-in
problem might highlight the necessity of policies, aimed at disseminating greater
information on quality among consumers.
To our knowledge this is the first article that studies analytically the property of a model
with multiple goods, two-side uncertainty and heterogeneous agents. Moreover, we
endogenize the number of firms, i.e. we discuss entry, and in addition we discuss failure,
i.e. exit. This means that this paper is launching the basis for an overall treatment of the
establishment of a new market under heterogeneity and interaction.
Further research includes the use of more sophisticated firms (oligopoly or conjectural
variations models), and the characterization of the welfare properties along the continuum
of equilibria. Of course the model could be simulated to study the shapes of alternative
diffusion curves, and to enquire how final outcomes depend on initial conditions. As
regards this last point, we claim that our model can be a workhorse for scholars aiming at
simulating the diffusion patterns of multiple goods considering explicitly the role of
supply and not only of demand. This setup is sufficiently flexible to account for a network
topology of the updating process (indeed, an agent might receive information from a
subset of agents), and for many kinds of noisy disturbances. An interesting aspect of the
story is that rationality of agents populating these artificial worlds is not so bounded as to
abandon Bayesian decision theory altogether.
12
References
Abrahamson E. and Rosenkopf L. (1993), Institutional and competitive bandwagons: using
mathematical modelling as a tool to explore innovation diffusion, Acad Manag Rev, 18,
487-517
Agliardi E. (1998), Positive Feedback Economies, London, Macmillan
Amable B. (1992), Effets d’apprentissage, compétitivité hors-prix et croissance cumulative, Econ
Appl, vol. 45(3), 5-31
Aoki M. and Yoshikawa H. (2002), Demand Saturation-Creation and Economic Growth, J Econ
Behav Org, 48, 127-154
Arthur B. W. (1994), Increasing Returns and Path Dependence in the Economy, Ann Arbor:
University of Michigan Press
Bass, F. (1969), A new product growth model for consumer durables, Manag Sci, 15, 215-227
Bergemann, D. and Välimäki J. (1997) Market diffusion with two-sided learning, Rand J Econ,
28(4), 773-795
Bertoletti, P., Fumagalli E. and Poletti C. (2008) Price-Increasing Monopolistic Competition? The
Case of IES Preferences. IEFE Working Paper No. 15. http://ssrn.com/abstract=1303537
Bikhchandani S., Hirshleifer D., and Welch I. (1992) A Theory of Fads, Fashion, Custom, and
Cultural Change as Informational Cascades, J Political Econ, 100(5), 992-1026
Bogliacino, F. and Rampa G. (2008) Quality Risk Aversion, Conjectures, and New Product
Diffusion, Working Paper, University of Genoa, http://ssrn.com/abstract=1136922
Chatterjee R. and Eliashberg J. (1990), The innovation diffusion process in a heterogeneous
population: a micromodeling approach, Manag Sci, 36, 1057-1079
De Groot M. H. (1970), Optimal Statistical Decision, New York, McGraw Hill
Dixit A. K. and J.E. Stiglitz (1977) Monopolistic competition and optimum product diversity, Am
Econ Rev 67, 297-308.
Duffie D. and Manso G. (2006), Information Percolation in Large Markets, Am Econ Rev, 97(2):
203–209
Feder G. and O’Mara G.T. (1982), On information and innovation diffusion: a Bayesian approach,
Am J Agric Econ, 64, 145-147
Geroski, P. A. (2000), Models of technology diffusion, Res Policy, 29, 603-625.
Granovetter, M. (1978), Threshold models of collective behavior, Am J Sociol, 83, 1420-1443.
Grimmett G. (1999), Percolation, 2nd edition, New York, Springer
Hall, B. H. (2005) Innnovation and Diffusion, in J. Fagerberg, D. C. Mowery and R. R. Nelson,
The Oxford Handbook of Innovation, pag 459-485
Jensen R. (1982), Adoption and diffusion of an innovation of uncertain profitability, J Econ
Theory, 27, 182-193
Lancaster P. and Tismenetsky M. (1985) The Theory of Matrices. Second edition with
applications, San Diego; Academic Press
Lopez Pintado D. and Watts D. J. (2006) Social Influence, Binary Decisions and Collective
Dynamics, Working Paper, Institute for Social and Economic Research and Policy,
Columbia University.
Macy, M. (1991), Chains of cooperation: threshold effects in collective action, Am Sociol Rev,
56, 730-747.
13
McKenzie, L. W. (2002) Classical General Equilibrium Theory. Cambdridge, Massachusetts: MIT
Press.
Nelson, R. R., Peterhansl, A. and Sampat, B. (2004) Why and how innovations get adopted: a tale
of four models, Ind Corp Change, 13(5), 679-699
Rampa, G. (1989) Conjectures, Learning, and Equilibria in Monopolistic Competition, J Econ,
49(2), 139-163
Roberts, J. H. and Urban, G. L. (1988) Modelling Multi-attribute Utility, Risk, and Belief
Dynamics for New Consumer Durable Brand Choice, Manag Sci, 34(2), 167-185
Tirole, J. (1988) Industrial Organization. Cambridge, MA: MIT Press
Tsur Y., Sternberg M. and Ochman E. (1990), Dynamic modelling of innovation process adoption
with risk aversion and learning, Oxford Econ Pap, 42, 336-355
Valente, T. W. (1996), Social network thresholds in the diffusion of innovations, Soc Netw 18,
69-89.
Vettas, N. (1998) Demands and Supply in New Markets: Diffusion with Bilateral Learning, Rand
J Econ, 29(1), 215-233
Verbrugge R. (2000), Risk Aversion, Learning Spillovers, and Path-dependent Economic Growth,
Econ Letters, 68, 197-202
Wilkinson, J. H. (1965) The algebraic eigenvalue problem, Oxford: Oxford Science Publications
Yanagita, T. and Onozaki T. (2008) Dynamics of a market with heterogeneous learning agents, J
Econ Interact Coord, 3(1), 107-118
Young P. (2005) The Spread of Innovation through Social Learning, CSED working paper
12/2005
Young P. (2007) Innovation Diffusion in Heterogeneous population, Economic Series Working
Papers, 303, University of Oxford, Department of Economics
14
Appendix
A.1. Proof of Proposition 3
The system is highly non-linear, so we should limit ourselves to discuss local stability, using a
linear approximation in a neighbourhood of one equilibrium. The Jacobian matrix of y t = F (y t −1 )
is easily checked to be the following one:
  ∂µ t 


  ∂µ t −1 
 0 nM ,nM

J =   ∂m t 
  ∂µ 
  t −1 
0
 4 n ,nM


 ∂µ t 
0 nM , 4 n 

 0 nM , 2 n
 ∂τ t −1 

I nM
0 nM , 2 n
0 nM , 4 n 

 ∂m t   ∂m t   ∂m t  

 
 

 ∂τ t −1   ∂m t −1   ∂γ t −1  
 ∂γ t 
0 4 n ,nM 
I 4 n 

∂
m
 t −1 

(12)
where I k is the k-identity matrix and 0 k , p is a k-by-p null matrix.
The stability condition is that all the eigenvalues of J, evaluated at an equilibrium, do not lie
outside the unit circle. We need some preliminary results.
Claim 1. At an equilibrium, the eigenvalues of J are those of the four blocks along its main
diagonal.
Proof. We need simply to prove that
[Γ

 1


′
+
x
x
=
Γ
+
i ,t −1
i ,t i ,t
 i ,t −1 − p
 i ,t

]
∂m t
= 0 . Define first
∂γ t −1
− pi ,t  
  ≡ A i ,t (Γ i ,t −1 , m i ,t −1 )
2 
p i ,t  
(13)
and
[x
D
i ,t (Qi ,t − x′i ,t m i ,t −1 )
where g i (⋅) ≡ [g i (⋅)
]
1 0   (QiD,t − x′i ,t m i ,t −1 ) 


=
0 pt  − (Q D − x′ m )


i ,t
i ,t i ,t −1 
≡ B i ,t (m i ,t −1 )g i (m i ,t −1 , µ t −1 , τ t −1 )
(14)
′
− g i (⋅)] and g i (⋅) was defined before expression (11). Define finally
Ci ,t (Γ i ,t −1 , m i ,t −1 ) ≡ [ A i ,t (Γ i ,t −1 , m i ,t −1 )]−1 B i ,t (m i ,t −1 )
(15)
Summing up, firm i’s updating formula can be written as
m i ,t = m i ,t −1 + Ci ,t (Γ i ,t −1 , m i ,t −1 )g i (m i ,t −1 )
(16)
and the block which interests us now is:
15
  ∂m1,t     ∂C1,t 
 ∂g1  

  
 g1 + C1,t 

∂γ
∂γ
 ∂γ t −1  
 ∂m t    t −1     t −1 

M

= M =

 ∂γ t −1    ∂m     ∂C 
 ∂g n  
n ,t 
n ,t


g + C n ,t 

  ∂γ     ∂γ  n
 ∂γ t −1  
  t −1     t −1 
(17)
which is clearly equal to zero, since from (11) g i (⋅) = 0 in equilibrium, and the g i ’s themselves do
not depend on firms’ precisions.
QED
We can thus concentrate on the four principal blocks of J. The NW block has eigenvalues lower
than one, and tending to one as time goes to infinity: they are the weights attached to consumers’
prior means in the updating formulae: see (3) above. The second and fourth blocks give rise to
respectively nM and 4n eigenvalues equal to one: they relate to the updating of consumers’ and
firms’ precisions, and are immaterial for stability. In fact changes in the precisions do not affect
the equilibrium itself, being more important in the initial, rather than the final, phases of the
learning process (Rampa, 1989).
 ∂m t 
We are thus left with the 2n eigenvalues of the block 
.
 ∂m t −1 
 ∂m t 
Claim 2. The eigenvalues of 
 are as follows:
 ∂m t −1 
(i) n eigenvalues are equal to one, implied by the continuum of equilibria;
(ii) the other n eigenvalues are equal to ρ(Dt G t ) + 1 , where ρ(⋅) is the column vector of the
eigenvalues of the argument, 1 is a column vector of ones, Dt is a diagonal matrix with positive
diagonal elements, and G t is a matrix with positive extra-diagonal elements.
Proof
(i) Define the following matrix:
C1,t

Ct =  0


 0
0 

... 0 


0 C n ,t 
(18)
g = [g1 (m t −1 )′ ... g n (m t −1 )′]′ .
(19)
0
and
Given (16), and given that g i (⋅) = 0 in equilibrium, one deduces:
 ∂m t 
 ∂g 

 = I 2 n + Ct 

 ∂m t −1 
 ∂m t −1 
(20)
This matrix has 2n eigenvalues equal to 1 plus those of the second term. Since by construction
 ∂g 
g (⋅) is formed by 2n terms, n of which are the opposite of the remaining n, 
 has rank n,
 ∂m t −1 
 ∂g 
and the same is generically true for Ct 
 : hence the latter has n eigenvalues equal to zero.
 ∂m t −1 
16
 ∂m t 
Thus, we can conclude that n eigenvalues of 
 are unitary. These n unitary eigenvalues
 ∂m t −1 
correspond precisely to the very existence of the n-dimensional continuum of equilibria: a move
along this continuum is followed neither by divergence nor by convergence to the previous point.
This completes the proof of part (i) of Claim 2.
 ∂g 
(ii) In order to study the remaining n eigenvalues of Ct 
 , we can write:
 ∂m t −1 
 ∂g   ∂g   ∂p t 

=


 ∂m t −1   ∂p t   ∂m t −1 
(21)
 ∂g 
where 
 is 2n by n, and
 ∂p t 
 ∂p t 

 is n by 2n, and p t = [ p1,t ... pn ,t ]' .
 ∂m t −1 
 ∂g   ∂p t 
It is well known that the non-zero eigenvalues of Ct 

 are the same as those of
 ∂p t   ∂m t −1 
 ∂p t   ∂g 
∂p t
has the following expression:

Ct 
 . Exploiting (5),
∂m t −1
 ∂m t −1   ∂p t 
 ∂p t 


 ∂m t −1 
 ∂p1
 ∂α
 1
 0
=

 M

 0

∂p1
∂β1
1
β
 1

1 0
= 
2 M


0

− α1
0
0
M
∂p2
∂α 2
M
∂p2
∂β 2
M
0
0
0
0
L L
0
L L
0
O O
M
L L
∂pn
∂α n
0
0
L L
0
1
− α2
β2
β 22
L L
0
M
M
M
O O
M
0
0
0
L L
1
β12
0
βn

0 

0 


M 
∂pn 

∂β n 
(22)

0 

0 


M 
− αn 

β n2 
Each of the diagonal blocks of matrix C t , in turn, can be written18 as:
 γ i,2
+ tpi ,t
Ci ,t = d i ,t  pi ,t

t
where d i ,t =
18

tpi ,t 

γ i ,1 + t 
pi ,t
(
γ i ,1γ i , 2 + t γ i , 2 + γ i ,1 pi2,t
(23)
) > 0 for t < ∞ , and recalling that γ
See Bogliacino and Rampa (2008), expression C.2 of the Appendix.
17
i,k
≡ γ i , k ,0 , k = 1,2 .
 ∂g 
Hence, the product Ct 
 is easily seen to be a 2n-by-n matrix composed by n-by-n column
 ∂m t −1 
vectors of the following form:
 γ i,2 

 ∂g i
d i ,t  pi ,t 
− γ  ∂pk ,t
 i ,1 
(24)
 ∂p t   ∂g 
As a result of (22)-(24), the product 
Ct 
 is a n-by-n matrix with elements:
 ∂m t −1   ∂p t 
 ∂g
d
α
1  γ i 2
d i ,t
+ i2,t γ i1  i = i ,t
2  β i1,t pi ,t β i ,t  ∂pk ,t 2 β i1,t
 γ i 2 α i ,t  ∂g i


 p + β γ i1  ∂p
i ,t
 i ,t
 k ,t
(25)
We can thus write the expression:
 ∂p t   ∂g 
Ct 

=
 ∂m t −1   ∂p t 
 d1,t  γ 1, 2 α1,t


+
γ 1,1  L


 2β1,1,t  p1,t β1,t

=
M
O

d n ,t

0
L

2 β n1,t

0
M
 γ n, 2 α n,t

p +β
n ,t
 n,t
  ∂g
 1
  ∂p1,t
 M

  ∂g n
γ n,1  
  ∂p1,t
∂g1 

∂pn,t 
O
M  ≡ Dt G t

∂g n 
...
∂pn,t 
...
(26)
where Dt is a diagonal matrix with positive diagonal elements.
 ∂g 
We need now to prove that the extra-diagonal elements of the matrix G t ≡  i  are positive.
 ∂pk ,t 
Using the definition of g i (⋅) after expression (14), one can see that the elements outside the main
diagonal of G t are the derivatives of the demand w.r.t. prices of the other goods (
∂x i
∂pk
, i ≠ k ).
Remind that the consumer’s problem is:
n
max
∑
n
∑px
u ( xi ) f (λi )s.t.
i =1
i i
≤w
(27)
i =1
whose first order condition is (calling z the Lagrange multiplier):
u ' ( xi ) f (λi ) − zpi = 0 .
If we calculate
(28)
∂xi
∂xi
pk
∂z
using the implicit function theorem, we get
, where
=
∂pk
∂pk u ' ' ( xi ) f (λi ) ∂pk
∂z
is defined by the boundary condition:
∂pk

n
∑ p u'
i
i =1

p z 
 i   − w = 0
 f (λi )  
−1 
(29)
Thus we have:
18
 pz 
'  p z 
pz 
u '−1  i  + i  u '−1  k 
∂z
 f (λi )  f (λk ) 
 f (λk ) 
=−
n
∂pk
pi z  −1 '  pi z 

 u ' 

 f (λi ) 
i =1 f (λk ) 
[ ]
(30)
[ ]
∑
Simple manipulation of the numerator above, in particular using the Inverse Function Theorem,
shows that the numerator itself is negative as long as:
( )
u ' xi*
>1
*
*
u ' ' xi xi
(31)
( )
where xi* is a solution to (28). With additively separable preferences the LHS is nothing else than
the elasticity of demand (Bertoletti et al., 2008), so (31) is certainly satisfied with our
∂xi
formulation19 u (x ) = xδ . So (30) is negative, and we can conclude that
> 0, i ≠ k . This
∂pk
completes the proof of Claim 2.
QED
Claim 3. If at an equilibrium the elasticity of the conjectured demand is greater than the elasticity
of the true demand, Dt G t has at least one positive eigenvalue.
Proof
As we said, the elements of Dt are positive for t < ∞ . From the definition of g i (⋅) and from (26)
it follows that the i-th element along the main diagonal of G t can be written as
∂g i ∂xi
=
∂pi ∂pi
−
TrueDemand
∂xi
∂pi
(32)
ConjecturedDemand
By definition of elasticity, using the fact that at an equilibrium the price-quantity couple is the
same for the true and the conjectured demand, the assumption of Claim 3 is equivalent to
∂xi
∂pi
−
TrueDemand
∂xi
∂pi
< 0.
(33)
ConjecturedDemand
Since both true and conjectured demand are negatively sloped, (33) implies that (32) is positive.
Using the results of Claim 2, part (ii), the fact that (29) is positive, and finally the fact that Dt is a
positive diagonal matrix, we conclude that all elements of Dt G t are positive. Claim 3 then
follows from the Perron-Frobenius Theorem20.
QED
We can finally complete the proof of Proposition 2. From Claim 3 and Claim 2, part (ii), it
 ∂m t 
follows that 
 has an eigenvalue greater than one. Claim 1 says that the eigenvalues of
 ∂m t −1 
 ∂m t 

 are also eigenvalues of J; hence J has an eigenvalue greater than one.
 ∂m t −1 
QED
19
Indeed one can argue that the property is completely general, since a firm will never find optimal to fix a
price where the elasticity of demand is lower than one. However, this condition is true only for conjectured
demand, and not for the true one.
20
See Lancaster-Tismenetsky (1985), Theorem 1 on page 536.
19
A.2 Proof of Proposition 4
Using Claim 2, part (ii), we need to prove that Dt G t can have a real eigenvalue lower than −1. In
what follows we will drop the time subscripts for easiness of notation, writing DG instead of
Dt G t : in fact we are evaluating the jacobian matrix J at an equilibrium (all consumers have
converged to the true quality value and prices are fixed at the equilibrium values). We start once
more from some preliminary results.
Claim 4. The following statements hold:
i) G is symmetric, and one has G = G1 + G 2 , where G1 is diagonal and G 2 has rank 1;
ii) DG has the same eigenvalues as D1/ 2GD1/ 2 ;
iii) D1 / 2 GD1/ 2 = DG1 + D1 / 2 G 2 D1 / 2 .
Proof
(i) To find the elements of matrix G, we differentiate the equilibrium conditions (11) w.r.t. prices,
using the definition of demand (3) and imposing equilibrium condition. We get:
 ∂g i

 ∂pi

 ∂g i
 ∂pk
=
1
QiD + β i +
pi (δ − 1)
1 δQiD D
=
Q >0
(1 − δ ) Mw k
1 δQiD D
Q
(1 − δ ) Mw i
i=k
(34)
i≠k
which implies that G is symmetric21.
Hence, G = G1 + G 2 : the first matrix is a diagonal matrix with elements
1
pi (δ − 1)
QiD + β i , and
δ
1
QQ′ , where Q is the vector of equilibrium quantities. But the non-zero
1 − δ Mw
1
δ
eigenvalues of G 2 are the same as those of
Q' Q , so G 2 has rank 1.
1 − δ Mw
(ii) We know from Claim 2, part ii, that D is diagonal and non singular, thus it admits D1/ 2 . But
G2 =
DG is similar to D −1 / 2 (DG )D1/ 2 = D1/ 2GD1/ 2 , and the two matrices have the same eigenvalues.
(iii) Given part (i) above, and since diagonal matrices commute, one can write
D1 / 2 GD1 / 2 = D1 / 2 G 1 D1 / 2 + D1 / 2 G 2 D1 / 2 = DG 1 + D1 / 2 G 2 D1 / 2
where the first term is a diagonal matrix, while the second term is
product of D1/ 2Q and itself, hence is symmetric.
21
The
(35)
δ
1
times the external
1 − δ Mw
QED
∂g i
, i ≠ k , are equal to the extra-diagonal terms of the Jacobian of the demand functions, thus in
∂pk
the general case symmetry holds only for compensated demands (see Theorem 1: McKenzie, 2002, p. 10),
and not for the Marshallian ones, because of income effects (McKenzie, 2002, p. 12). However the
condition holds under our present assumptions.
20
Define now the following terms: φi , as the i-th element of the main diagonal of D; ν i ≡
the ratio between firm i’s initial precisions; and si ≡ ε T
γ i1
γ i 2 , as
ε i , as the ratio between the true elasticity
and the conjectured elasticity in the market for product i. We proceed with the following
Claim 5. The following statements hold:

1
1 + ν i pi2 1 + 
1
 εi  ,
(i) the eigenvalues of DG1 are (1 − si )
2 γ i1 + t 1 + ν i pi2
(
(ii) the non zero eigenvalue of D1 / 2G 2 D1/ 2 is
)
1
δ
2
i = 1, K , n


1 

1 + ν k pk2 1 +  
 pk Qk
 εk  
sk


pk Qk
γ k1 + t 1 + ν k pk2 

 k



∑ ∑
k
(36)
(
)
(37)
Proof
(i) We concentrate first on the i-diagonal element of the diagonal matrix D, φi . From the
αi

α
βi
d γ
1
definitions after (23) we have φi = i  i 2 + i γ i1  =
. From (5) one
2
β
2 β i  pi β i
2
i γ i1γ i 2 + t γ i 2 + γ i1 pi

α
α
deduces in addition that i = 2 pi − ci , hence one can write pi i = pi2 + ( pi − ci ) pi . But the
βi
βi
γ i 2 + γ i1 pi
(
)
monopoly pricing rule (Tirole, 1988) is that the Lerner Index is equal to the inverse of the
α
p
p2
conjectured elasticity, whence ( pi − ci ) = i ; then we have pi i = pi2 + i . Given the
βi
εi
εi

1
1 + ν i pi2 1 + 
γ
1
 εi  .
definition ν i ≡ i1 , we get thus φi =
2β i γ i1 + t 1 + ν i pi2
γ i2
(
)
Now, recall from the proof of Claim 4, part (i), that the i-diagonal element of the diagonal matrix
p
1
G1 is
Qi + β i . In equilibrium one has Qi = ( pi − ci )β i , which is equal to i β i by the
pi (δ − 1)
εi
monopoly pricing rule. Using the true elasticity ε T =
1
pi (δ − 1)
1
ε
and the definition si ≡ T , we obtain
εi
1−δ
Qi + β i = β i (1 − si ) .
The i-th diagonal element of the diagonal matrix DG1 , that is its i-th eigenvalue, is then equal to
β i (1 − si )φi . Substituting the value of φi found above, we get (36).
1 1/ 2
δ
D QQ′D1/ 2 , a matrix that has rank 1 and
1 − δ Mw
thus a single non-zero eigenvalue. It is easily checked that this matrix has the same non-zero
δ 1
δ 1
eigenvalue as,
Q′DQ =
φk Qk2 . Exploit again the equilibrium fact
1 − δ Mw
1 − δ Mw k
(ii) Using Claim 4, we write D1 / 2 G 2 D1/ 2 =
∑
21
Qk = ( pk − ck )β k , the monopoly pricing rule
( pk − ck ) =
pk
εk
, and the definitions sk ≡
εT
and
εk
1
1
, which together imply
Qk2 = Qk pk sk β k . Substituting finally the value of φi found
1−δ
1−δ
in part (i) above, and using the budget constraint Mw =
pk Qk , one gets (37).
QED
εT =
∑
k
We will use the following result, which we call Claim 6.
Claim 6. If A and B are two symmetric matrices and if rank(B) = 1 , then the i-th eigenvalue of
A+B, say ρ i ( A + B) , is equal to ρ i ( A ) + mi ⋅ ρ (B ) , where mi∈ [0,1] , and ρ (B ) is the only nonzero eigenvalue of B.
Proof. See Wilkinson (1965), pp. 97-98.
QED
We are finally ready to complete the proof of Proposition 4, stating that an eigenvalue of J, say
ρ i (J ) , can be lower than −1. By Claim 2, part (ii), this means ρ i (DG ) < −2 . We will posit
sufficient conditions for this result.
We know from Claim 4 that DG has the same eigenvalues as D1 / 2 GD1/ 2 , and that
D1 / 2 GD1/ 2 = DG1 + D1 / 2 G 2 D1 / 2 , the sum of two symmetric matrices. Claim 5, in turn, gives
expressions for the eigenvalues of DG1 and D1 / 2 G 2 D1/ 2 . Claim 6 asserts finally that ρ i (DG ) =
(
)
(
)
ρ i (DG1 ) + mi ⋅ ρ D1/ 2G 2 D1/ 2 , with mi∈ [0, 1] , implying ρ i (DG ) ≤ ρi (DG1 ) + ρ D1/ 2G 2 D1/ 2 .
Suppose now that the true elasticity ε T is very high, implying δ ≈ 1 (recall that δ < 1 anyway),
and that all firms but the i-th one conjecture an elasticity ε k very near to the true one, while the ith firm conjectures a low elasticity ε i . This implies sk ≈ 1 and 1 ≈ 0 for all k ≠ i ; at the same
εk
time, the i-th firm will price very high, so that (2) implies a low share of consumers’ expenditure
on good i; in addition, si is very high. Suppose further that all firms have low initial precisions of
their α parameters, so that γ k1 is near to zero, ∀k . Finally, consider the system at the very start of
the learning process, meaning t = 1 . Looking carefully at (37), all this implies that
ρ D1/ 2G 2 D1/ 2 ≈ 1 2 .

1
1 + ν i pi2 1 + 
ε
1
1
i 

This given, Claim 6 ca be written as ρ i (DG ) ≤ (1 − si )
for the i-th
+
2
2
1 + ν i pi2
eigenvalue of DG.


1
1
1 + ν i pi2 1 + 
1 + ν i pi2 1 + 
1
 ε i  < −5 . This
 ε i  + 1 < −2 , meaning (1 − s )
We need to have (1 − si )
i
2
2
2
1 + ν i pi
1 + ν i pi2
might well be the case, given our current assumptions of a low ε i and a high si , and if in addition
one assumes that ν i is high, i.e. firm i is initially more uncertain on the β parameter than on the α
parameter. Notice that, as time passes ( t > 1 ) and hence γ k1 grows above zero, the result does not
hold any longer.
(
)
This completes the proof of Proposition 4.
QED
22
A.3 Proof of Proposition 5
Let introduce a fixed cost of entry equal to F. In order to calculate the optimal degree of
diversification, we need to fix price equal to marginal cost, introduce lump sum taxation for an
nF
nF
for each consumer (reducing her income to w −
) and maximize the indirect
amount
M
M
utility function in n. The demand for goods of identical quality is:
p1/(1−δ ) [δf (λ )]
M (w − n F M )
=
1/(1−δ )
δ /(1−δ )
np
np
[δf (λ )]
1/(1−δ )
Q D = M (w − n F M )
(38)
Replacing the price equal to marginal cost, the indirect utility function is given by
n1−δ c −δ M δ ( w − nF / M )δ , which must be maximized in n, considered as a real variable for
simplicity. The first order condition is also sufficient, due to the strict concavity of the indirect
utility function, and is the following:
n e = (1 − δ ) Mw / F =
Mw
Fε T
(39)
The equilibrium condition with endogenous number of firms is a zero profit condition for the
marginal firm, defined by the price p = min{ pi | pi = pi* } (where pi* are equilibrium prices),
given the equality of marginal cost and quality through firms (and convergence of consumers’
conjectures in equilibrium). Hence:
( p − c)Q = F
α − cβ Mw
=F
2β n p
(40)
By simple algebra we get
n* =
α − cβ Mw Mw
=
α + cβ F
Fε C
(41)
Over (respectively under) diversification is the case n* > n e (respectively n* < n e ). Replacing
with (39) and (41) completes the proof.
QED
23